They are waterproof.
Paramecia is one of the three Devil Fruit types. These fruits give the users a power that can affect their body, manipulate the environment, or produce substances.
There are three different types of Paramecia, and Doflamingo seems to be the third type. He can produce strings, rather than his body being made of strings. We have clearly see him get hurt and bleed and additionally have we seen strings separated from his body, still being intact (eg: birdcage). The strings that he produces come out of his body and are thus not part of it any more. He can manipulate their shape and structure, and he can use his parasite technique to make them move, but they are just strings and nothing more.
Proof that they are indeed separate and do not affect him, has been given in chapter 787.
When Franky and the Tontatta try to push back the birdcage with the seastone factory, the strings are basically touching seastone, yet we don't see them nor Doflamingo get any weaker. So whether it be the birdcage or the string clones he makes, regardless of them being touched by water or seastone, they will not lose their power.
On a sidenote, it seems that this effect happens with every paramecia that produces things, except for Robin. I think, Robin would be affected if her hands touch the sea(stone). I'm basing this on the fact that she could be turned into a doll, by touching one of her limbs, while similarly it does not seem like Doflamgino would be turned into a doll if his cage or clone was touched.